# Category:Absolutely Continuous Functions

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This category contains results about absolutely continuous functions.

Let $I \subseteq \R$ be a real interval.

A real function $f: I \to \R$ is said to be **absolutely continuous** if it satisfies the following property:

- For every $\epsilon > 0$ there exists $\delta > 0$ such that the following property holds:

- For every finite set of disjoint closed real intervals $\closedint {a_1} {b_1}, \dotsc, \closedint {a_n} {b_n} \subseteq I$ such that:
- $\ds \sum_{i \mathop = 1}^n \size {b_i - a_i} < \delta$

- it holds that:
- $\ds \sum_{i \mathop = 1}^n \size {\map f {b_i} - \map f {a_i} } < \epsilon$

- For every finite set of disjoint closed real intervals $\closedint {a_1} {b_1}, \dotsc, \closedint {a_n} {b_n} \subseteq I$ such that:

## Pages in category "Absolutely Continuous Functions"

The following 10 pages are in this category, out of 10 total.